Smoothed boundary conditions, coherent low-frequency scatter, and boundary modes

Abstract

Low-frequency coherent scatter from a rough surface may be conveniently investigated using a linear boundary condition applied to a smoothed surface, of the form ∂φs/∂z=ηφ, where φ, φs are solutions of the wave equation representing, respectively, the total and scattered field potentials. The validity of the theories discussed here is restricted to kh≲1, where k is the wavenumber and h the mean spacing between roughness elements. The constant η is a function of frequency, angle of incidence, and type of roughness; in the general case of scatterers distributed isotropically over an interface between two fluids it is sensitive to eight physical parameters. Methods of calculating η for various types of rough boundary are discussed, and comparisons are made—notably between the boss and the stochastic perturbation models. Also examined are interesting implications of the smoothed boundary conditions; e.g., the boundary wave which is a true propagating mode corresponding to energy trapped in the vicinity of a rough surface, and which is only generated by a source near this surface (it thus differs fundamentally from the evanescent modes of a diffraction grating which may be excited by plane waves and are therefore not true boundary modes). For source and receiver near the boundary, and for negligible incoherent scatter, the farfield amplitude of the boundary wave may exceed that of the direct (normal) acoustic arrival—a fact which has been verified experimentally in model work. Incoherent scatter introduces an attenuation factor exp(−δr), where r is the source–receiver distance and δ is proportional to the sixth power of the frequency.